Theoretical and numerical issues concerning temporal stabilisability and detectability

Motivated by the design of perturbation (output) feedback controllers for nonlinear systems, temporal system properties have been developed by the authors, for time-varying linear continuous-time systems. In particular temporal controllability/reachability, temporal stabilisability, temporal reconstructability/observability and temporal detectability. As opposed to their ordinary counterparts, these temporal properties identify the temporal loss of stabilisability and detectability that may occur for time-varying linear continuous-time systems obtained e.g. by linearising around state and control trajectories of a non-linear system. One contribution of this paper is to show that temporal stabilisability and temporal detectability require a measure of state decay that cannot be independent of the state representation and state vector norm. This raises the issue of how to select the state representation or norm. The second contribution of this paper is to deal with this selection. Thirdly this paper presents two new, alternative ways to compute temporal stabilisability and detectability and their associated measures. They are compared with computations proposed earlier that rely on LQ control. Finally, through simple illustrative examples, numerical aspects concerning computation of temporal stabilisability and detectability and their associated measures are investigated.